Infinitesimal

Amir Alexander

Why I looked at this book

The notion of infinity is commonplace nowadays, but I can't help feeling that we accept it too readily. In particular, should we really accept the idea that a small volume of space contains an infinitude of points. I'm hoping that this book will give information on the arguments people had when such things were still a matter for debate.

First impressions

The start of the book seems to have more about theology and less about mathematics than I'd expected. We find out about Martin Luther and the start of Protestantism, and how the Pope and the Holy Roman Emperor Charles V had to decide how much to put aside their differences to combat it. I hope that there'll be more about the mathematics as the book goes on.

Main review

This is a book of two halves, the first concerning the Jesuit battle with those who promoted infinitesimals (including the followers of Galileo), the second the battle between Thomas Hobbes and John Wallis in England. One thing I note is that it hardly looks at what happened outside the particular time being considered - no questioning of how many angels can dance on the head of a pin, none of the later work to add rigor to the use of infinitesimals in the calculus. It's really more of a biographical work.

In the first half I felt that this didn't really work. We find out about the rise of the Jesuit order, at the expense of the (too similarly named) Jesuat order, but there are too many different people for this biographical format to work. Although there is a bit more of the mathematical thoughts than in the sample, I was still left wondering precisely what the contradictions were in infinitesimial mathematics that the Jesuits campaigned against.

The second half was much better. With just two main characters, the biographical format didn't have the same problems as in the first half. We find out how John Wallis and Thomas Hobbes struggled with the changing politics of the English civil war and restoration. Hobbes rose higher in the establishment, and claimed (much as the Jesuits did) that he wanted to keep dangerous ideas such as infinitesimals out of mathematics so as to have a secure bedrock on which to found his ideas. However since he also claimed to have squared the circle his mathematical ideas weren't taken that seriously. Wallis was involved with the nascent Royal Society, and I thought it interesting how they distrusted existing mathematics as being too much related to tradition-bound thought, and found the new infinitesimal ideas as more in keeping with their experimental ethos.

In conclusion, if it were just the second half, I would recommend this book as an informative and readable biographical work, even if it doesn't have much about the longer term progress of infinitesimal thought. For the full book I'm not so sure - the first half was a bit of a struggle.